We prove that the exponent of Ωm(G) is at most pm if G is a powerful p-group with p odd. Calling on a recent result of Héthelyi and Lévai, we prove that $ for all m. These results also hold for regular p-groups. We also bound the nilpotence class of a subgroup of a powerful group by e + 1, where pe is the exponent of the subgroup. This is just one more than what the bound would be if the subgroup were itself powerful.

We classify the finite 2-groups G whose integral group rings Z[G] have the multiplicative Jordan decomposition property. In addition to those cases already known, these include three further cases of order thirty-two and no others. We also give a theorem which severely restricts the structure of those finite groups G with Z[G] having this property which are not of order 2 a 3 b for some a, b.

We prove that a certain class of finite p-groups has the property that the product of pth powers is again a pth power.

No abstract

We discuss the relationship between the derived length and the number of character degrees in the restricted setting of a normally monomial p-group, G, of maximal class. We continue the Lie algebra approach implemented by Keller, Ragan, and Tims. With a number of technical results, we improve the existing bound, dl(G) ≤ 1 2 | cd(G)| + 11 2 , to obtain, dl(G) ≤ 2 5 | cd(G)| + 5.